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HOW TO FI.ND 



THE CHURCH FESTIVALS 



WITHOUT TABLES; 



BEING A LETTER ADDRESSED TO THE CHAIRMAN OF THE COMMITTEE 
ON THE PRAYER BOOK LN THE HOUSE OF CLERICAL AND 
LAY DEPUTIES OF THE GENERAL CONVEN- 
TION OF THE CHURCH, IN 1871. 






19k 

FREDERICK A. P." BARNARD, S.T.D., LL.D., 

ii 7 7 7 

President of Columbia College, New York City. 







M. H. MALLORY AND COMPANY, HARTFORD, CONN. 

1872. 






THE UBRARYf. 
OF CONGRESS jj 

WASHINGTON 



Entered, according to Act of Congress, in the year 1872, by 

F. A. P. BARNARD, 
In the Office of the Librarian of Congress, at Washington. 



LETTEK 



Columbia College, New York, 
April 5, 1871. 
Reverend and Dear Sir, m 

At your request I here repeat the substance of a communication which 
I ventured to address to you during the session of the General Convention of 
the Church in 1868. 

In the introduction to the Prayer Book there are given tables exhibiting 
the time of Easter for one or two cycles of the moon ; and also rules for 
computing the time when Easter will occur in years beyond the limits of 
the tables. These tables are very well, and serve the purposes of most 
persons. But for one who desires to know on what day Easter occurred in 
some year long past, or on what day Easter will occur in some future year 
not embraced in the tables of the Prayer Book, the rules of computation, as 
given, are not so simple as could be desired, nor are they easy of application 
without the use of pen or pencil. The object of my communication of 1868 
was to suggest the inquiry whether, in the new edition of the Prayer Book, 
it would not be advisable to introduce some simple rules, easily fixed in the 
memory, by means of which the place of Easter in the calendar of any 
year, past or future, may be found mentally, and that without any very 
painful effort. Such rules I have been accustomed myself, for a long time, 
to use ; and I furnish them herewith for your examination. 

The place of Easter depends, first, upon the place of the Paschal full 
moon, and secondly, upon the Dominical or Sunday Letter. 

The place of the Paschal full moon depends upon the Golden Number, 
or the place of the year in the lunar cycle ; and upon another term which, 
throughout an entire century, and sometimes throughout two or three entire 
centuries, is constantly the same, but which sometimes also changes by a 
unit in passing from century to century. This change may be called the 
Secular Correction. 

In order to find Easter we must, therefore, be able to find, first, the 
Golden Number ; secondly, the Sunday Letter ; and thirdly, the constant 
term, as modified by the Secular Correction. This latter is unimportant if 
the Easter sought falls within our century, since the constant for this cen- 
tury may be learned once for all. 

It may be observed, in general, in regard to the rules which follow, that 
the number denoting any given year of our Lord may be resolved into 
two parts, one of them being a certain number of complete centuries, and 
the other a certain number of years of an incomplete century. These two 
parts are treated separately, and the results combined. Thus,*1871 is divided 
into 18 and 71. 

1. To Find the Golden Number. 

Divide the years by twenty, and add the quotient to the remainder. 

Divide the centuries * by four, and add the quotient to five times the remain- 
der. 

Add together the two results, and increase the sum by one. If the final result is 
nineteen, or less, it is the Golden Number. If not, subtract nineteen, and the 
remainder is the Golden Number. 

Example.— What is the Golden Number in 1871 ? 18 -f- 4 = 4, with 2 
remainder. And 4 + 5x2=14; 71 -4- 20 = 3, with 11 remainder. And 
3 + 11 = 14. Finally, 14 + 14 + 1 = 29. And 29 - 19 = 10, Golden Num- 
ber. 

* If the centuries exceed nineteen, drop nineteen as often as possible before proceeding 
with this division. 



2. To Find the Sunday Letter. 

According to the rules of the Gregorian calendar, the centurial years 
which are exact multiples of 400 are leap years, and the three intermediate 
centurial years have only three hundred and sixty-five days each. In every 
succession of four centurial years, beginning with the leap year centurial, 
the Dominical Letters return in the following order, viz., A, C, E, G. Or 
giving (as is most convenient) to the letters the numerical values which cor- 
respond to their places in the alphabet, they present the series, 1, 3, 5, 7. 
This simple series is easily remembered. The number in this series which 
corresponds to the hundreds of a given year may then be called the centurial 
belonging to that date. 

If the years of the incomplete century be divided into twenties, and the 
excess of twenties resolved into fours, the Dominical Letter will advance 
three places for every twenty, two places for every four, and six places for 
every unit of the still outstanding remainder. Hence the Sunday Letter 
will be found by taking the sum of four numbers, which may be called the 
centurial, the vigesimal, the quaternial, and the residual. And hence the 
rule : 

In the series 1, 3, 5, 7, find the number corresponding to the given century {remember- 
ing that the number 1 belongs to the century divisible by four) and call this the cen- 
turial. 

Jlultiply the twenties of the incomplete century by three, and call the product the 

VIGESIMAL. 

Multiply tJie fours in the excess of twenties by two, and call the product the quater- 
nial. 

Multiply the final remainder by six, and call the product the residual. 

If the sum of the numbers thus obtained is seven, or less, it is t7ie numerical value of 
the Sunday Letter. If it b" greater than seven, subtract seven as often as may be 
necessary to reduce it to seven or below, and the final residt is the Sunday Letter.* 

Example. — What is the Sunday Letter of 1871 ? 

1st. For the centurial. The first century of the quadricentennium being 
16, the third is 18, of which the centurial is 5. 

2d. There are three twenties in 71, and 3x3=9= the vigesimal. 

3d. There are two fours in 11 (71-60), and 2x2 = 4= the quaternial. 

4th. The final remainder (or excess of fours in 11) is 3. and 3 x 6 = 18 
= the residual. 

Then 5 + 9 + 4 + 18 = 36, and 36 - 5 x 7 = 1 = A. 

Note. — That in summing up the terms, the sevens may be dropped dur- 
ing the operation, thus simplifying the solution: 9 + 5 = 14, for instance, 
being equal to twice seven, may be disregarded ; and from 18, fourteen may 
be dropped. The final summation will then be 4+4 = 8, and 8 — 7 = 1 = 
A. 

The foregoing rule may be kept in mind by means of the following for- 
mula, in which A stands for the value of the Dominical Letter. C, V, Q, 
and R are the initials of the terms employed above, and n denotes any 
number : 

A = C+3V + 2Q + 6R-7ti. 

Alternative Rule. — The following process, though involving occasion- 
ally larger numbers, may be found by some persons more easy of applica- 
tion. Let it first be understood that the difference between seven and any 
number less than seven, is called the complement of that number. 

Then, the centurial is found as before ; the vigesimal is discarded, and 
the quaternial is one half the number expressing the latest leap year in the 
incomplete century. Thus, for 1797, the centurial is three, and the quater- 
nial is one half of 96 (the latest leap year in 97) ; that is, it is 48. The resid- 

* To find the Sunday Letter in old style, the process is the same, except in respect to 
the centurial. The centurial for old style is found by adding three to the number of the 
complete centuries, and suppressing seven from the result as often as possible. The 
vigesimal, quaternial, and residual are the same for both old style and new. 



ital, as under the former rule, is one, and its complement is six. Add then 
the centurial, the quaternial, and the complement of the residual, viz., 3 + 
48 + 6 = 57 ; and suppress the sevens, viz., 7 x 8 = 56 ; and there remains 
1 = A = the Sunday Letter of 1797. Hence the concise rule : 

To the centurial add one half the number of the largest leap year in the incomplete 
century, and the complement of the years in excess of leap year. The sum suppressing 
sevens, is the value of the Dominical Letter. 

3. To Find the Value of the Constant* used in determining 
Paschal Full Moon, as modifled by the Secular Correction. 

From the number of the centuries take its fourth part and its third part {disregarding 
fractions in both cases), and increase the result by two. 

Example. — What is the constant for 1871 ? 

18 + 4 = 4. 18 -*- 3 = 6. And 18 - 4 - 6 + 2 = 10. 

The constant is the same for every year of the century to the 99th, but 
exclusive of the 100th. 

What is the constant for 2371 ? 

93 + 4 = 5. 23 + 3 = 7. And 23 - 5 - 7 + 2 = 13.f 

This rule is true up to the year 4200 ; but in that year, and those following, 
the number of the century must be diminished by one before taking the 
third part. In other respects the rule remains unaltered. And in the year 
6700, and subsequently, up to 9200, the number of the century must be 
diminished by two before taking the third part. In the year 9200, and subse- 
quently, up to the year 11,700, the rule will be the same as that given first 
above, only that the result is to be increased by three instead of two. But 
long before that time it is probable that the Gregorian calendar will be 
found itself to require correction. 

4. To Find the Place (in the Calendar) of the Paschal Full 

Moon for any Year. 

To four times the Golden Member add the constant for the century. If 
the Golden Number is even (but not otherwise), increase this sum by fifteen. Then, if 
the result is greater than twenty, and less than fifty, it is the date of tlxe Paschal 
full moon considered as a day of March. If it exceed thirty-one, subtract thirty- 
one, and the remainder is the date of Paschal full moon in April. 

If, however, the foregoing result be not greater than twenty, and less than fifty, 
add thirty or subtract thirty (or twice thirty, if necessary) to bring it icithin these 
limits. 

Note that if the number obtained by this process be exactly twenty, or 
exactly fifty, it cannot be brought within the limits by the addition or sub- 
traction of thirty. In this case the date of Paschal full moon is to be taken 
at forty-nine. 

Note, also — and particularly — that if the number obtained by the rule 
be itself forty-nine, the Golden Number being, at the same time, twelve, or 
more than twelve, the date of Paschal full moon is to be taken at forty- 
eight. 

Example. — What is the date of Paschal full moon in 1871 ? 

The Golden Number, as just found, is ten, and the constant for the cen- 
tury is ten also. The Golden Number, moreover, is even. Then, 
P = 10 x 4 + 10 + 15 = 65. And 65 - 30 = 35. 

Paschal full moon is, therefore, the 35th day of March ; that is, the 4th 
day of April. 

* What is the date of Paschal full moon in 2258 ? 

* This constant, for old style dates, is always two. The secular correction was intro- 
duced with the Gregorian calendar, the uncorrected constant being also increased from 
two to nine. 

t These numbers diminished by nine will give those corresponding to the same centu- 
ries which are found in Table II. of the general tables at the end of the introduction to the 
Prayer Book. 



6 

By the rules foregoing, we find the Golden Number to be seventeen, and 
the constant for the century twelve. Then, 

P = 17 x 4 + 12 = 80. And 80 - 30 = 50 * 

The date of Paschal full moon, in this case, must be taken, according to 
note first above, at 49. It is, accordingly, the 49th of March, or the 18th of 
A«pril. 

What is the date of Paschal full moon in 3966 ? 

The Golden Number is fifteen, and the constant for the century is nine- 
teen. Then, 

P = 19 + 4 x 15 = 79. And 79 - 30 = 49. 

But because the Golden Number exceeds eleven, this must be taken, 
according to the second note foregoing, at 48. Paschal full moon is, there- 
fore, the 48th of March, or the 17th of April. 

5.. To Find Easter Sunday. 

To the constant number, eighteen, add the value of the Sunday Letter, and after- 
ward, if necessary, add seven, or such number of sevens as may suffice to make the 
sum greater than the date of Paschal full moon considered as a day of March, and no 
more. This sum is the date of Easter considered as a day of March. If it exceeds 
thirty-one, thirty -one is to be subtracted, and the remainder is the date of Easter in 
April. 

Example. — What is the date of Easter in 1871 ? 

We have found the Sunday Letter to be A = 1, and the date of Paschal 
full moon to be 35. Hence, 

E = 18 + 1 + 7x3 = 40th day of March = 9th day of April. 

6. Miscellaneous Rules fob the Movable Feasts and Fasts op 

the Chubch. 

For Septuagesima Sunday. — Take four from the date of Easter (in leap 
year three) and go back two months. 

Example. — In 1871, Easter is April 9. 

April 9 — 4 is April 5, and Septuagesima Sunday is February 5. 

For Ash-Wednesday. — Add thirteen to the date of Easter {fourteen in 
leap year), and go back two months. 

Note. — If the sum exceed the number of days of the month in which 
Easter falls, consider it, nevertheless, a day of that month. 

Examples. — In 1871, Easter is April 9. 

April 9 + 13 = April 22 ; and Ash- Wednesday is February 22. 

In 1868 (leap year), Easter was April 12. 

April 12 + 14 = April 26 ; and Ash-Wednesday was February 26. 

In 1869, Easter was March 28. 

March 28 + 13 = 41 ; and Ash- Wednesday was January 41 = February 10. 

For Trinity Sunday. — Take five from the date of Easter, and go forward 
two months. 

Example. — In 1871, Easter is April 9. 

April 9 — 5 = April 4 ; and Trinity Sunday is June 4. 

For Whitsun-Day. — Take twelve from the date of Easter, and go forward 
two months. 

Example.— In 1871, Easter is April 9 = March 40. 

March 40 — 12 = March 28 ; and Whitsun-Day is May 28. 

* If, as in this example, the Golden Number is fifteen, or more than 15, the process may 
be simplified by dropping 15 before multiplying. Thus, 17 — 15 = 2. Then, 
P = 2x4 + 12 = 20. And 20 + 30 = 50, 

In the example next following the Golden Number is 15 ; and 15 - 15 = 0. Hence, when 
the Golden Number is 15, the constant for the centnry, or this constant increased cr dimin- 
ished bv 30 <*ive9 the date of Paschal full moon directly. Thus, in the example above, 

P= 19 + 30= 49; 
which, because 15 exceeds 11, must be put = 48. 



For Ascension-Day. — Take twenty-two from the date of Easter, and go 
forward two months. 

Example. — Easter in 1871 = March 40. 

March 40 — 22 = March 18 ; and Ascension-Day is May 18. 

For Advent Sunday. — The First Sunday in Advent can never be earlier 
than the 27th November, nor later than the 3d December = 33d November. 
Hence, 

To or from the date of Easter, considered as a day of March, add or sub- 
tract sevens till the result falls between the limits twenty-six and thirty-four, 
exclusive of both ; and the result is the date of Advent Sunday, considered 
as a day of November. 

Example. — Easter in 1871 is March 40. 

March 40 — 7 = March 33 ; and Advent Sunday is November 33 = De- 
cember 3. 

Otherwise. — To twenty-five (the date of Christmas-Day) add the value of 
the Sunday Letter (making A = 8 instead of 1, the other letters retaining 
their usual values) and the result is the date of Advent Sunday, considered 
as a day of November. 

Example.— The Sunday Letter in 1871 is A = 8 ; and 25 + 8 = 33d No- 
vember = 3d December. 

In 1870 the Sunday Letter was B = 2 ; and 25 + 2 = 27th November = 
Advent Sunday in 1870. 

It is convenient to be able to fix some points in the series of Sundays 
after Trinity. The smallest number of Sundays that can follow Trinity 
Sunday is twenty-two. 

For the twenty-second Sunday after Trinity, take four from the date of 
Easter, and go forward seven months. 

Example. — Easter Sunday in 1871 being April 9, we have 

April 9 — 4 = April 5 ; and the twenty-second Sunday after Trinity is 
November 5. 

The ninth Sunday after Trinity is found by taking three from the date 
of Easter, and advancing four months. Thus, in 1871, April 9 — 3 = April 
6 ; and August 6 is the ninth Sunday after Trinity. 

Hence, Trinity Sunday, the twenty-second Sunday after Trinity, and the 
ninth Sunday after Trinity, form this series, easily remembered : 

Easter — 5 = Trinity Sunday. 

Easter — 4 = twenty-second Sunday after Trinity. 

Easter — 3 = ninth Sunday after Trinity. 

The month to be supplied will be inferred from the numbers. 

The fifth Sunday after Trinity has the same date as Easter, three 
months later, when Easter is in April ; its date is one less when Easter is 
in March. 

The fourteenth Sunday after Trinity has the date of Easter increased by 
one, five months later * 

* The following will be found, perhaps, more curious than useful , 

The twenty-seventh Sunday after Trinity (which marks the largest number of Sundays 
which can happen between Trinity and Adyent), when referred to November, always hap- 
pens on the same day of the month as Easter Sunday referred to March. Thus Easter is 
the 40th day of March (the 9th of April), and the twenty-seventh Sunday after Trinity is 
the 40th day of November (the 10th day of December) in 1S71. November is also the eighth 
month after March. If now we add 13 to 27, continuously, suppressing 35 whenever the 
sum exceeds that number, adding also 3 to the month whenever 13 is added to the weeks, 
and subtracting 8 from the months whenever 35 is subtracted from the weeks, a series of 
Sundays will be obtained, of which the date will be successively owe less, two less, three 
less, and so on, than the date of Easter. 

Also, if we subtract 13 from 27, continuously, adding 35 whenever the subtraction is 
otherwise impossible, at the same time subtracting 3 from the months for every 13 from 
the weeks, and adding 8 to the months for every 35 added to the weeks, we shall obtain a 
series of Sundavs of which the dates are one greater, two greater, three greater, and so on, 
than the date of Easter. But it is to be noted in making these additions and subtractions, 
that if a result is at any time reached exceeding 29. and less than 35, this is to be disre- 
garded, and 13 added or subtracted again, the double addition or subtraction still changing 
the date only one unit. 



8 

To find the Number of the Sundays after Trinity.— From the whole 
number of days in March and April united {sixty-one) take the date of 
Easter as a day of March, and divide this difference by seven. The quotient 
is the number of Sundays to be added to twenty-two (the minimum number) 
in order to obtain the whole number of Sundays after Trinity. 

Note that, in April, the date may be taken directly from thirty. 

Example.— In 1871, Easter is April 9 ; and 30 - 9 = 21, which, divided 
by seven, gives three. 22 + 3 = 25, the number of Sundays between Trinity 
and Advent in 1871. 

To find the Number of Sundays after Epiphany. — From the date of 
Easter, considered as a day in March, take eleven {ten in leap year), and 
divide the result by seven. The quotient is the number of Sundays between 
Epiphany and Septuagesima Sunday. 

Example.— In 1871, Easter is March 40. 

40 — 11 = 29 ; and 29 •*- 7 = 4, the number of Sundays after Epiphany in 
1871. 

In 1872 (leap year) Easter is March 31 ; and 31 — 10 = 21, which, divided 
bv seven, gives three. Hence, there are three Sundays after Epiphany in 
1872. 



The foregoing rules are given without demonstration. It is proper, 
however, to present the reasons on which they are founded ; and this I will 
endeavor to do as succinctly as possible. 

I. As to the rule for the Golden Number. This is merely an arithmeti- 
cal artifice for performing with facility, and without the use of the pencil, 
the operation prescribed for the same purpose in the first of the tables of 
the Prayer Book. The first year of our era is historically ascertained to 
have been the second of the lunar cycle. Hence, the year of our Lord, in- 
creased by one and divided by nineteen (the number of years in a complete 
cycle), will leave as a remainder the number (in the cycle still incomplete) 
of the year under consideration. 

To divide by nineteen is troublesome. But if we consider that 100 con- 
tains five nineteens and five over, and 400 contains twenty nineteens and 
twenty over — that is to say, twenty-one nineteens and one over — we shall 
see that the cycle advances but one unit in 400 years. This gives the 
reason for dividing the hundreds by four, and taking the quotient as the 
first term of the result. Each hundred of the remainder still outstanding 
advances the cycle five places. Hence, the remainder is multiplied by five 
for the second term. The years of the incomplete century are easily 
resolved into twenties ; and each twenty is obviously one complete cycle 

These rules will hold true for a series of dates extending from twelve days before 
Easter to twenty days after. 

The following table embraces a portion of the dates which may be thus found. The 
letter E stands for the date of Easter, considered as a day of March. There are added the 
corresponding dates for the year 1872 : 

Date Months Corresponding 

No. of Sunday. of Sunday. after March. Dates in 1872. 

Twenty-sixth Sunday after Trinity E — 7 8 Nov. 24. 

Thirteenth " " E — 6 5 Aug. 25. 

Trinity Sunday E — 5 2 May 26. 

Twenty-second Sunday after Trinity E — 4 7 Oct. 27. , 

Ninth " " .... E — 3 4 July 28. 

Eighteenth " " .... E — 2 6 Sept. 29. 

Fifih " " .... E — 1 3 June 30. 

Twenty-seventh " " ... E 8 Nov. 31 = Dec. 1 

Fourteenth " " .... E + l 5 Aug. *32 = Sept. 1 

First " " .... E + 2 2 May 33 = June 2 

Twenty-third ll " ... E + 3 7 Oct. 34 = Nov. 3 

Tenth " " .... E + 4 4 July 35 = Aug. 4 

Nineteenth " " .... E + 5 6 Sept. 36 = Oct. 6 

Sixth * " .... E + 6 3 June 37 = July 7 

Twenty-eighth " " .... E + 7 8 Nov. 38 = Dec. 8 



9 

and one over. Each single year remaining 1 advances the cycle one also. 
Thus the second part of the rule is explained. To the united results thus 
obtained we add a unit, on account of the year of the cycle which had 
elapsed at the beginning of the era. Inasmuch as one hundred complete 
cycles amount to 1900 years, it is obvious that we may, before proceeding, 
reject nineteen from the number of the centuries as often as it occurs. 
Tims, if the number of the given year is very large, as 8963, we may reject 
four times nineteen, or seventy-six, from eighty-nine, leaving thirteen cen- 
turies to operate upon. In this case, also, we may adopt the simple mode 
above given of dividing by nineteen ; that is to say, we may divide first by 
twenty, and then add the quotient to the remainder. Thus : 
89 -J- 20 = 4, with 9 remainder ; and 4 + 9 = 13. 

Then, 13 -=- 4 = 3, with 1 remainder ; and 3 + 1 x 5 = 8. Also, 63 -=- 20 
= 3, with 3 remainder ; and 3 + 3=6. 

8 + 6 + 1 = 15, the Golden Number in the year 8963. 

II. In regard to the rule for the Sunday Letter. Here the first object is 
to find a simple way of disposing of the centuries. The common year 
begins and ends on the same day of the week. In the succession of ordinary 
years, therefore, the Dominical Letter goes backward each year one place. 
A leap year sets it back two places. Accordingly, in one hundred Julian 
years, the letter goes back one hundred and twenty-five places. But, if from 
this number we suppress the even sevens, we shall have a remainder of six 
only, showing that in a Julian century the Dominical Letter goes back six 
places, which is equivalent to a forward movement of one place. In the 
Gregorian calendar three centuries out of four have but twenty-four leap 
years instead of twenty-five. In each of these centuries the retrograde 
movement is, therefore, one less than that just found ; that is to say, is only 
five, which is equivalent to a forward movement of two. Hence, in every 
Gregorian quadricentennium there will be three centurial forward, steps of 
two places each, and one forward step of one place ; or, in all, an advance of 
seven places, completing the cycle of the letters. 

To find now the actual value of the letter for a given centurial year, Ave 
take the fact, historically ascertained, that in England, in 1752, after the in- 
tercalation in February (which intercalation really took place in 1751 in old 
style, as the year 1752 only began in Great Britain on the 25th March), the 
Dominical Letter was D. By the adoption, in that year, of the Gregorian 
calendar, the fourteenth day of September (which regularly fell on Monday) 
was put in place of the third of the same month (which was a Thursday), 
the day of the month being removed backward a week and four days. The 
letters daring the remainder of the year fell, therefore, as if all the days 
of the year preceding this change (including the first of January) had 
been removed, in like manner, backward four days in the week. And 
as January 1st fell on Wednesday (since the Sunday Letter was E 
before the intercalation of February), the removal of this day backward 
four places would have carried it to Saturday. In other "words, the 
effect of the suppression of eleven days in the calendar changed the 
Dominical Letters of the year from E and D to B and A* Between 1752 and 

* Valuable, and even necessary, as was the improvement made upon the Julian calendar 
by Pope Gregory XHT., it is impossible, at the present day, to regard without astonish- 
ment the extraordinary and totally unnecessary interference with systematic chronology 
caused by his arbitrary obliteration of ten days out of the month of October in the year 
153J. Julius Ccssar had reason for his "year of .confusion ;" for the calendar was then out 
of joint with the year by nearly three months. Moreover, his momentary disturbance of 
the order of things was only a more signal but final example of the chronic confusion 
which, through the aeency of the pontifical jugglers, had reigned before. Neither of these 
reasons existed in sufficient force to justify Pope Gregory in the introduction of his new 
year of confusion in 1582. The second did not exist at all; for the Julian calendar had 
secured a perfectly unbroken uniformity in the method of computing time for more 
than sixteen centuries. And the displacement of the seasons by the error of the Julian 
intercalation had been too trivial to occasion the slightest inconvenience, or to call for 
correction. What was desirable and all that was desirable, was to prevent any further 
displacement for the future ; and this was accomplished by the simple and easily intelli- 
gible correction of the intercalatron introduced by the mathematicians of Pope Gregory. 



10 

the close of the century, there elapsed forty-eight years, of which only 
eleven were leap years (the centurial years 1700, 1800, and 1900 being made 
common years by the Gregorian reformation). The Sunday Letter, there- 
fore, went back fifty-nine places during this period; or, suppressing the 
sevens, three places only, which is equivalent to advancing four places. 
And four places added to 1 (= A) carries us to 5 (= E), which was the 
value of the Dominical Letter in 1800. In 1900 the value will be found to 
have advanced two places more, so that it will have become 7 (= G). But 
in 2000 (a leap year) there will have occurred an advance of only one addi- 
tional place, giving a value of 8. And 8 — 7 = 1 = A, which is always the 
Sunday Letter of the bissextile centurials. Thus, starting with the centurial 
year divisible by 400, we have the successive centurial years of each quadri- 
centennium marked by the series, 1, 3, 5, 7 ; or A, C, E, G. 

In dividing the incomplete century into twenties and fours, the object is 
to obtain small numbers to operate on. and numbers into which the entire 
number is easily resolved. Every twenty years, from the beginning of the 
century, must contain five leap years (except, three fourths of the time, the 
last twenty ; but with this last one we have nothing at present to do). In 
every twenty years, therefore, the Sunday Letter moves backward twenty- 
five places ; which, suppressing sevens, is four places ; being equivalent to 
a forward movement of three places. 

In every additional four years the letter moves backward Jive places, 
which is equivalent to a forward movement of two places. 

And in every additional single year the letter moves backward one place, 
which is equivalent to a forward movement of six places. These outstand- 
ing single years can never be leap years, as is obvious from the fact that the 
last year of every twenty, and the last year of every four, previously taken, 
is a leap year. 

The reason of the rule for the Dominical Letter is thus made obvious. 
After a little familiarity with its use, it will be found more convenient, in 
dealing with the residual years, to subtract their number (it can never exceed 
three) from the numbers previously found, instead of adding six times their 
number ; or, otherwise, to subtract their number from seven, and add the dif- 
ference. In the first instance, however, it facilitates recollection, as it also 
gives uniformity of character to the rule throughout, to make the terms all 

The real reason for the suppression of the ten days was not that society was suffering or 
astronomy embarrassed in consequence of the inconsiderable retrogradation of the equi- 
noxes in March and September which had taken place daring the lapse of several centu- 
ries. It was that, in 1575, the vernal equinox fell on the 11th of March, whereas, in 325, at 
the time of the assembling of the Council of Nice, which put an end to the differences 
previously existing between the East and the West in regard to the time of celebrating 
the festival of Easter, it was supposed to have fallen on the 21st. That the restoration of 
this coincidence was of no practical importance, even in an ecclesiastical point of view, is 
manifest from the fact that, in the authorized Explicatio Romani Cafendarii, etc., pub- 
lished at Rome by the Pope's principal mathematician, Clavius, in 1603, the author takes 
pains to insist that the Church is under no obligatiou to make Easter a movable feast at 
all ; but that she does so merely out of respect to an ancient custom. Considering this 
fact, it is the more surprising that the extraordinary interruption in the succession of the 
days of the year introduced along with the new calendar should have been admitted at all, 
since it was by no means done without reflection, nor without extensive consultation with 
civil as well as with ecclesiastical rulers. As early as 1577, five years before the appear- 
ance of the Papal decree declaring the change, the whole scheme was submitted to all the 
Roman Catholic princes in Europe ; and it not only elicited no objection, but failed to pro- 
voke a remark of even doubtful approval. On the other hand, according to Montucla, it 
was everywhere eulogized in the most unqualified terms. 

Had it not been for this feature of the proposed reform, the old mode of computing the 
lapse of time would have passed into the new without any sensible dislocation at any point 
of the record. And whether or not it had been immediately received by peoples not in 
communion with the Church of Rome, it could have introduced no difference into the civil 
calendars of the East and the West, or of Romanists and Protestants, until at least the 
lapse of nearly a century and a quarter (the year 1700), long before which time it would 
probably have been universally accepted upon its own merits and without regard to the 
manner of its orijrin. These ten days undoubtedly prevented its acceptance in Protestant 
Ensland for nearly two hundred years (when the difference had become eleven days), 
and they still keep the great empire of Russia out of harmony in regard to this matter with 
the rest of Europe and with all America, the difference being now twelve days. 



11 

immediately additive. They will all be small, and the process by which, they 
are obtained may soon be so mastered as to make the determination of the 
Sunday Letter for any year in the past or the future a problem requiring but 
a moment's thought, without the use of any implements of calculation. 

In regard to the alternative rule, after what has been said above the ex- 
planation will be obvious. For every four years euding with a leap year, 
the Sunday Letter advances two places. By dividing the year from the 
beginning of the century by four, and multiplying the quotient by two, 
therefore, we shall have the total advance of the letter at the last leap year 
preceding the year given ; or in the given year itself, if that is a leap year. 
But this division by four and multiplication by two gives a result which 
may be more directly obtained by simply dividing by two at once. More- 
over, to add the complement of the residual is equivalent to subtracting the 
residual itself. Hence, it is seen that the alternative rule rests on the same 
principles precisely as the rule first given. 

The method may be applied to the finding of the Sunday Letters in the 
period when the old style of reckoning still prevailed, if we consider that, 
as all the centurial years were then leap years, the movement of the letter 
from one centurial year to another was always one place forward. Accord- 
ingly, after every seven centuries, the centurial year encounters the same 
letter a second time. It remains to ascertain the value of this letter for some 
known centurial year. We have seen that, in 1752, after the intercalation, 
the Sunday Letter was D. In the fifty-two years which had elapsed since 
1700, the Sunday Letter had gone back sixty-five places. Therefore, if we 
descend in time from 1752 to 1700, the letter should go forward sixty-five 
places from D (or 4), which we find to be the place in the year first men- 
tioned. 

65 + 4 = 69 ; and 69 -j- 7 = 9, and 6 remainder. 

Thus, the value of the Sunday Letter was 6 = F, in the centurial year 
1700, after the intercalation of that year. As by the Julian reckoning the 
letter advancs one place per century, it follows that if we go back seventeen 
places from F = 6, we shall find the letter corresponding to the year Zero, 
that is, to the year last preceding our era. Seventeen, with the sevens sup- 
pressed, becomes three, which, taken from six, leaves 3 = C for the Sunday 
Letter of the year 0. In the year one, therefore, the letter was 3 — 1 = 2 = 
B ; or, in other words, the era commenced on a Saturday.* 

* This result is obtained by computing backward, as above, to tbe beginning of the 
era, according to the Julian calendar as in use in the sixth century, and since, in the 
Western Churches. But according to Blondel (Histoire clu Calendrier, cited by Delani- 
bre) the differences between the Julian calendars of Alexandria and Rome occasioned 
lone disputes, which were threatening to lead to violence, when happily the parties were 
pacified by the successful efforts of Dionisius Exiguus, a monk of Rome, who persuaded 
the Western Christians to adopt the Alexandrian calendar. What probably contributed 
largely to his success was his proposition that all Christians should unite in referring dates 
to the year of our Saviour's birth. Before this time, 'says Delambre {Histoire de V Astro- 
nomie Moderne, vol. i.), there had been, in this respect, no uniformity of usage, some 
counting the years from the era of Diocletian, which they called also the Era of the Mar- 
tyrs; some from the day ot the Passion ; some, like the Romans, irom the foundation of 
Rome ; and others designating years by the names of the consuls or emperors. 

The first year of the Julian calendar, according to Montucla, Delambre, and other 
authorities, corresponded to the forty-sixth before our era. By a misunderstanding on 
the part of the priests, to whom was committed the charge of the calendar, the intercala- 
tion was made twelve times successively at the end of every third year, instead of every 
four; the period of four years being made out, according to a method of reckoning com- 
mon among the Romans and in the East, by counting the bissextile year at the beginning as 
well as that at the end. There had then been introduced three intercalary days too many ; 
and to correct the error thus occasioned, it was ordered by Augustus that the intercalation 
should be omitted during the twelve years beginning with the thirty-seventh of the era, and 
ending with the forty-eighth. If these numbers are to be depended upoa, it would seem 
that, in the settlement of the controversy among the early Christians in regard to the 
Julian calendar, the regularity of the succession of bissextile years, in the order in which it 
was established by its founder, was interrupted : a not unnatural consequence of the yield- 
ing of Rome (where the true order was most likely to be preserved), for th« sake of peace, 
to Alexandria. However this maybe, it is evident thatany interpretation of the figures will 
make it probable that one intercalation, and only one, was suppressed subsequently to the 
commencement of our era. The effect of this, if it occurred, was to move backward the 



12 

The years 100, 200, 300, etc., were marked by letters advanced one place, 
two places, three places, and so on, beyond C ; that is to say, their letters may 
be found by adding three to the number of the century. If the number 
exceed seven, seven must, of course, be suppressed. Hence, ' for years 
reckoned according to the old style, the Sunday Letters are found in the same 
way as for Gregorian years, except that the centurial term is not to be taken 
from the series heretofore given, viz , 1, 3, 5, 7, but ascertained by adding 
three to the number of the century, and suppressing sevens. 

III. In order to explain the rule for the secular correction, and for the 
Paschal full moon (IV.), it is necessary first to say a word in regard to the 
relation between the lunar month and the solar year, on which the correc- 
tion depends. The periods of revolution of the earth and moon around 
their respective centres of motion are not commensurable ; nor are they 
connected by any approximate numerical relation which is at all obvious. 
The discovery, therefore, which was made in the fifth century before our 
era, that there is a period, at the end of which these two bodies return to 
their original positions relatively to the sun, with a very inconsiderable 
difference, was a very interesting, and in some respects important, incident 
in the history of astronomy. This period is nineteen years, and the near- 
ness of approach to coincidence of the two movements at its close will be 
understood from the following statements : 

The length of the true solar (tropical) year is 365 days, 5 hours, 48 
minutes, 46.05444 seconds ; or, expressed in days and decimals, 365.2431997. 
The length of a mean lunation is 29 days, 12 hours, 44 minutes, 2.84 sec- 
onds ; or 29.530588259 days. Hence, nineteen tropical years contain 
6939.6017943 days, and two hundred and thirty-five mean lunations, 
6939.688.2801 days. The difference is 0.0864858 days, amounting to a little 
more than two hours four and a half minutes, by which the lunar period is in 
excess. Supposing, therefore, that at the commencement of a given year 
the moon is at a certain determinate point of advancement in the lunation, 
then at the beginning of the twentieth year following (or the end of the 
nineteenth) it will not be quite so far advanced ; or, in common language, 
its age will be less. The age of the moon at the beginning of the year is 
what is called its epact ; and it thus appears that the effect of the inequal- 
ity above spoken of is to occasion a slow diminution of the moon's epact 
from cycle to cycle. It is this slow diminution which necessitates the appli- 
cation, to the constant employed in finding the date of the Paschal full 
moon, of the secular correction which we are considering. As the epact 
diminishes, the time of a particular phase of the moon (as, for instance, the 
new or the full moon) is retarded in its occurrence, or pushed forward to a 
later period in the month. The retardation amounts to a day in about two 
hundred and twenty years (219.7), and to thirty days, or the length of an 
entire lunation, in 6,600 years. In the second of the general tables relating 
to the calendar in the Prayer Book, are found the val ues of the necessary 
corrections corresponding to the successive centuries, as computed in the 
sixteenth century by the mathematicians of Pope Gregory XIII. Sixty-six 
hundred years added to the fifteen centuries preceding the Gregorian 
reform, carry us to the eighty-first century. The correction in the table is 
twenty-eight, instead of thirty. This arises from the fact that the length of 
the tropical year, and of the mean lunar month, as now received, differ 
slightly from those employed in the original calculation of the table. Ac- 
cording to Clavius, the authorized expositor of the reformed calendar, the 
diminution of the moon's epact amounted to only 0.080825237 days per 
cycle ; at which rate a retardation to the extent of an entire lunation would 
require the lapse of something over seven thousand years. Seven thousand 
added to fifteen hundred gives 8,500, opposite to which we find in the table 

Dominical Letter of the first year of The era a single place, or irom B to A ; so that, in point 
of fact, this year began on Sunday instead of on Saturday, as it is computed to have done 
above, and as it would have done had the intercalations from the beginning been regularly 
made. 



13 

the correction 30, or 0. The error of Clavius, if corrected, would produce 
no difference in the tabular numbers before the year 4800, and therefore 
need not concern us now. 

An inspection of the numbers in the table will show that they do not 
increase uniformly, so as to keep pace with the uniform diminution of the 
epact. The reason of this is, that, for practical purposes, it is necessary that 
the civil year should be made to consist of a determinate number of entire 
clays ; and, therefore, as the astronomical year contains a fraction of a day, 
the civil years cannot all be equal in length. It is also convenient to make 
the lunar months, in the calendar, to consist of entire days, which renders it 
similarly necessary to make the successive months unequal. The several 
steps by which Clavius effected the adjustment of the lunar to the solar 
period were the following : 

As a first approximation, the solar year was taken at 365 days exactly. 
A lunar year was then assumed of twelve months, consisting alternately of 
thirty-days and of twenty-nine days each. This gives, in effect, a mean lunar 
month of twenty-nine and a half days (which is about three quarters of an 
hour too short), and a lunar year of 354 days, less than the solar year by 
eleven days. If a calendar new moon occurs, therefore, at the beginning of 
a year exactly, it is obvious that, at the beginning of the next year, there 
will be a calendar moon eleven days old. In other words, the epact of the 
moon is annually increased eleven days. Supposing the original epact to be 
Zero, there will then be, in three years, an epact of thirty-three days. As 
this exceeds a lunation, a lunation may be dropped, and the epact corres- 
pondingly reduced. After three years more it will be necessary to do the 
same thing again ; and in the course of a cycle this necessity will arise 
seven times. The lunations thus superadded to the regular lunar months 
are called embolismic, which term is expressive of superaddition. The em- 
bolismic months are all made to consist of thirty days, except the last one of 
the cycle, which has but twenty-nine days. The reason of this will appear 
from the following comparisons, which are given for a period of four com- 
plete cycles, or seventy-six years : 

76 years of 365 days each, contain 27,740.000 days.* 

940 true mean lunations " 27,758.753 " 

Difference 18.753 " 

These 18f days are a little more than balanced by the nineteen intercalary 
days belonging to the leap years in tlie four lunar cycles, of which no 
account has yet been taken. But 940 lunations, taken at twenty-nine and 
a half days each, amount to only 27,730 days, or fall short by ten days of the 
number found in seventy-six common years. As there are twenty-eight 
embolismic months in the four cycles, it is possible to dispose of these ten 
days by adding half a day each to twenty out of the number, making them 
thus to consist of thirty instead of twenty-nine and a half each. There 
remain, then, eight months of mean length, four of which may be reduced 
to twenty-nine days, while the other four are made thirty. Of the twenty- 
eight embolismic months, therefore, twenty-four, or six in each cycle, will 
have thirty days ; and four, or one in each cycle, will have twenty-nine days. 
For convenience, this short embolismic month is put last. 

But we see now that the intercalation has put the solar period in excess 
by about a quarter of a day in seventy-six years ; to that extent increasing 
the moon's epact. In four times 76 years, that is, in 304 years, this 
increase will amount to a day. If the more exact numbers be taken, it will 
appear that nearly 308 years are necessary to increase the epact an entire 
day. The numbers employed by Clavius gave 312£ years. This is equiva- 
lent to eight days in 2,500 years. He proposed, therefore, to increase the 
epact by a day at the end of every 300 years, seven times successively ; and 
to make the addition of an eighth day at the end of the following 400 years. 
This is the method of making the lunar correction actually employed in the 
Gregorian calendar. But this correction adjusts the place of the moon in 



14 

tlie Julian year. The error of the Julian rear was provided for in the new- 
calendar, by a correction also taking effect in passing from century to 
century. The excess of the true year above 365 days is not quite a quarter 
of a day, but a fraction expressed by the decimal 0.2421997. The Julian 
calendar adds one day to every fourth year, or 100 days in 400 years. But 
if we multiply the foregoing fraction by 400, the result will give'us 96.87988 
days, — that is, not quite 97. The Julian calendar, therefore, adds about 
three days too many in every four hundred years. The Gregorian correction 
consisted, accordingly, in reducing three leap years in every four centuries to 
common years : and for convenience and facility of recollecting the correc- 
tion, the years chosen to be so reduced were the final years of those centu- 
ries which are not divisible by four ; those which are so divisible continuing 
to be leap years. It is evident from the numbers above given that this cor- 
rection is not quite adequate. There remains outstanding a minute error, 
which may amount to a day in something more than 3,300 years. Clavius 
supposed it to be greatly less than this, and regarded it, therefore, as too 
unimportant to require attention. At least, he proposed to leave it to pos- 
terity to look after. 

As this solar correction makes the year begin sooner than it otherwise 
would, it reduces the age of the moon, or the epact, by the same amount. 
The lunar and solar corrections are, accordingly, opposed* to each other ; but 
the solar, being the greater on the whole, prevails, and the epact gradually 
diminishes. As the lunar correction is applied once in three centuries, and 
the solar three times in every four, the two corrections may occasionally 
come together ; in which case the epact stands over unaltered from one cen- 
tury to the next. If the lunar correction falls at the end of a quadricenten- 
nium, the epact goes a unit forward. This would happen regularly once in 
1200 years, and no oftener, if the interval between the successive lunar cor- 
rections were always three centuries, and no more. But as the correction 
stands over to the fourth once in twenty-five centuries, the recurrence of 
this advance is not quite regular. Usually, when there is a change, the 
movement is retrograde. The numbers in the General Table II., in the 
Prayer Book, are increasing ; but, as related to the epact, they are sub- 
tractive. 

From what has been said, it will be manifest that the moon of the eccle- 
siastical calendar has very little to do with the actual moon in the 
heavens ; agreeing with that only in the length of its mean period. It was, 
in fact, a part of the deliberate design of the contriver that the ecclesiastical 
moon should always fall at least a day later in the month (although it does 
not always happen so) than the true moon, in order that the festival of 
Easter might not fall upon the same day on which the Jews celebrated their 
Passover, or the Quartadeciman Christians their Easter. He claimed for it, 
in fact, no astronomical significance, but called it what it is, simply a cycle. 
His manner of indicating epacts in the actual calendar of the year shows 
his independent disregard of the real moon. In the first place, the days of 
the year are divided off into twelve lunar months of thirty and twenty-nine 
days each, alternately ; the divisions falling as it may happen among the 
days of the ordinary calendar months. The end of the "lunar year thus falls 
on the 20th of December. Xo modification of this mode of division is made 
on account of the additional day of leap year ; the 29th of February being 
counted with the 28th as one day* The eleven days outstanding at the end 
of the lunar year constitute the increase of the epact in passing to the year 
following. Next, it is assumed that, whatever may be the age of the moon 
on the 1st of January, the same will be true of the successive moons 
throughout the year, at the beginning of the successive lunar months. And 

* This is in accordance with modern municipal law. Bat. in point of fact, in the Church 
Calendar, it was the 25th, which was counted as one day with the 24th ; this latter being the 
dies sextus ante Kalenda* Martias, and the leap year 25th, that which was later known as 
the dies bissextilis ante Kal. 2far., which gave name to the year. 



15 

as, by counting backward from January 1st into December a number of days 
equal to the epact, we shall come to the place of the new moon next pre- 
ceding January 1st, so by counting back from the first day of the second 
lunar month (which is January 31st) we shall find the place of the new 
moon in January, and so on. In order to save the trouble of counting, and 
to enable a person to find the place of new moon from the epact by inspec- 
tion, the contriver of the calendar marked in, in Roman numerals, the epacts 
corresponding to all the days of new moon possible in a month ; these 
numbers forming a series descending from thirty to one; thirty standing 
opposite the first day, and one opposite the last day of the lunar month. 

In the application of this method there arises an embarrassment, grow- 
ing out of the fact that, since the alternate lunar months have only twenty- 
nine days, they have not places enough for thirty epacts. This difficulty is 
surmounted by the expedient of writing, in the calendar of these months, 
two consecutive epacts opposite to one day, which is equivalent to bringing, 
occasionally, in the short months (or, as they were called, the hollow months) 
of the same cycle, two new moons to the same day, when they fell on dis- 
tinct days in the long or full months. The epacts XXIV and XXV were 
selected for this duplication, the reason of which choice 'will presently 
appear. In the following table is presented the arrangement of the calen- 
dar according to Clavius, for the first four months of the year, which, as it 
extends beyond the latest date of Easter, is sufficient for our present pur- 
pose: 



4 


Jajtuabt. 


M 


Febbttaby. 


4 


Mabch. 




£ 


APErL. 




a 






i a 






c 






a 






s 












o 






o 






M 


















O 


i 




It 


i 




t— 1 
o 
>> 






o 


ca 


1 




o. 


"5 


a 


ft 


V 


c; 


ft 


o 


a 


ft 




P 


K 


i-3 


P 


w 


A 


P 


W 


A 


P 


W 


Hi 


1 


or XXX. 


A 


1 


XXIX. 


D 


1 


or XXX. 


D 


1 


XXIX. 


G 


2 


XXIX. 


B 


2 


XXVIII. 


E 


2 


XXIX. 


E 


2 


XXVIII. 


A 


3 


XXVIII. 


C 


3 


XXVII. 


F 


3 


xxvin. 


F 


3 


XXVII. 


B 


4 


XXVII. 


D 


4 


XXVI., 25 
XXIV.XXV 


G 


4 


XXVII. 


G 


4 


XXVI .,25 
XXIV.XXV 


c 


5 


XXVI. 


E 


5 


A 


5 


XXVI. 


A 


5 


D 


6 


XXV., 23 
XXIV. 


F 


! 6 


XXIII. 


B 


6 


XXV., 25 
XXIV. 


B 


6 


XXIII. 


E 


7 


G 


7 


XXII. 


C 


7 


C 


7 


XXII. 


F 


8 


XXIII. 


A 


8 


XXI. 


D 


8 


XXIII. 


D 


8 


XXI. 


G 


9 


XXII. 


B 


9 


XX. 


E 


9 


XXII. 


E 


9 


XX. 


A 


10 


XXI. 


C 


10 


XIX. 


F 


10 


XXI. 


F 


10 


XIX. 


B 


11 


XX. 


D 


11 


XVIII. 


G 


11 


XX. 


G 


11 


XVIII. 


C 


12 


XIX. 


E 


12 


XVII. 


A 


12 


XIX. 


A 


12 


XVII. 


D 


13 


XVIII. 


F 


! 13 


XVI. 


B 


13 


XVIII. 


B 


13 


XVT. 


E 


14 


XVII. 


G 


1 14 


XV. 


C 


14 


XVII. 


C 


14 


XV. 


F 


15 


XVI. 


A 


15 


XIV. 


D 


15 


XVI. 


D 


15 


XIV. 


G 


16 


XV. 


B 


16 


XIII. 


E 


16 


XV. 


E 


16 


xni. 


A 


17 


XIV. 


C 


17 


XII. 


F 


17 


xrv. 


F 


17 


XII. 


B 


18 


XIII. 


D 


18 


XI. 


G 


18 


XIII. 


G 


18 


XI. 


C 


19 


XII. 


E 


19 


X. 


A 


19 


XII. 


A 


19 


X. 


D 


20 


XI. 


F 


20 


IX. 


B 


20 


XI. 


B 


20 


IX. 


E 


21 


X. 


G 


21 


VIII. 


C 


21 


X. 


C 


21 


VIII. 


F 


2-2 


IX. 


A 


22 


VII. 


D 


22 


IX. 


D 


22 


VII. 


G 


23 


VIII. 


B 


23 


VI. 


E 


23 


VIII. 


E 


23 


VI. 


A 


24 


VII. 


C 


24 


V. 


F 


24 


VII. 


F 


24 


V. 


B 


25 


VT. 


D 


! 25 


IV. 


G 


25 


VI. 


G 


25 


IV. 


C 


26 


V. 


E 


26 


III. 


A 


26 


V. 


A 


26 


III. 


D 


27 


IV. 


F 


27 


II. 


B 


27 


IV. 


B 


27 


II. 


E 


23 


III. 


G 


28 


I. 


C 


23 


III. 


C 


28 


I. 


F 


29 


II. 


A 








29 


II. 


D 


29 


or XXX. 


G 


30 


I. 


B 








30 


I. 


E 


30 


XXIX. 


A 


31 


or XXX. 


C 








31 


or XXX. 


F 









The conditions of the problem in regard to Easter are these : Easter 
must be a Sunday, and it must fall later than the full moon which happens 
on or next following the vernal equinox. But the words full moon are not 
to be understood to mean the astronomical full moon, either true or mean ; 
but the fourteenth day of an ecclesiastical lunation, of which lunation the 



16 

degree of advancement, is expressed ay die enact of the year. In tins court 
of fourteen da ys, the day of the new moon is itself htriodrd ; so that the 
date of full moon, so far as concerns the determination of Easter, will he 
ascertained by deducting thirteen from the epaet of the year (adding tAirty, 
if necessary, to make the subtraction possible), and g» t * J M»g the calendar 
-:1 :'_f ~'ii'.: 7i_~ ::e ::i:; ::' "_?~: :s nir Ft-ih :*_:?". iz. rreis-ri :~ 
thirty, if we deduct thirteen, there remains twenty-six, and this number in 
-_t ilr-iir :iii? :::.s;:r "It "■.': i-~ :: Mi::!' izi :ie -1:"_ ii- :i ±zrl 
lite first of these dates, faffing before the equinox (March 21), is rejected, 
and the other, April 4th, is taken as the date of Paschal toll moon. Now, as 
the Sunday Letter of the year is A, we look down the column of letters after 
April 4 until we come to A, which stands opposite April 9, and 
:'_i: :: :~ :"-- ii'.r :i Zi?:- f r Ivl. 

It was supposed by Clavius, in accordance with the 
edge of his time, that the vernal equinox would fall generally on the 20th, 
and never later than the 21st of March. This, therefore, determined the 
:-r :: -"-f r^lLr-r; -ossil^ Zis^r f-Z 



moon, being the fourte e nt h day earlier, or the 21st less thirteen days, is 

'::■: :r_: - :l :: :Lt -'_ ::' Mi::L :;-.:-.:-:: -~'_::i - ;ie :-.:".■= r.-,z.zs ."_- 



enact XXIII For any larger enact, as X.XIV, the 

be looked for a month later, or in April. Snce, therefore, the epaet JUL111 

the earnest ne^ i:-:: on wLi;i Easier can depend, so* the epaet 
" r. _ ef :Lt Li: en -l~.i :"_« 17:^1:5 :: L--r ":•—"- :"_f 
mining the »irtfrff r of the calendar to bring together, in the hollow 
the two enacts XXIT and XXV, rather than any two others, opposite to 
the same day. The effect of this is to : 

7th uf yLs^L bo consist uf Twerrr-iire dsr 5 only, while all ! 
r_ri:-r :- :'_- iiv 5 :: Mini, eir^r :_i~ :"_- 7:1 zo-sis; : 
I: Lls:~~i£e- :: roffil'.T :li: -.-: ::~ -:•:-- - :"_t nz 
occur on the same day of April, via, the 5th; and this is 1 
which, howerer uniniporiant it may appear, Clavii 
he ought to guard. He therefore resorted 10 an expedie 
"t zzziirTS-.Z'-.'i :j ::zi.irT^z :'zr tT^::? :: :"_r ?- : ■■r-essi-r -f.£' 
as they stood in the century in which he lived, and subseque 
formation of the calendar. They are as follows, the nnrnbera 
JtZz? :: :1t :~:lf l-ri^r ;:li:ei :~rr :~_e~ n-zTi'lj 

L i. 4, 

I XH XT7TT ~ 

1*. la. M 1: 

n. m. xxit. v. 



IT 




TIL 




a 


la. 

STL 


17. 




X1Z 






__t T-i": :- :ir «.tv:J line ri:~-i :'-:-sr i":-:-t :-fn ir. :'zz zyr.. :j 1 
single unit in each case. The yaar nanaagr in the second line exceeds that in 
the first, in each case, by efesen. We perceive the reason of this law 
we consider that, in eleven rears, the epaet is i ncreased eleven time 
sarely by eleven each time, or by 131 in all; and that out of tins 
:: :: Til":-: '„;-:: ~:-:_: :. :_-.r:j ii~i :::'_ ire ;:::;-: ^i": 1 
der of only one. In the whole aeries of nineteen epacts there can, 
ingly, be found eight near* of numbers differing in each case by a unit; but 

:i--:: ":e ::n: :z::: ill :"--r -.i^tt- ^: nil; ^ - 
which, put together, will form a continuous 

r'.-i::= ire rrii~i.ll 7 L:zz:zl zs'zz-i :~ :1: ijil: 
---. -'--r : - :-z- -J.'. :.: ..1_.-l.-7i - :i".> iii *._f: 
uncha nged. It will ha ppen, ti a gefo r c, that 
XXT are in the series, XXVI cannot be in it at the same t ime. JBya dnmg, 
for example, a unit to all the epacts giv en abov e, XXIT andXXV would 
belong to the 3d and 14th ye an; but XX VL which stands under 6 at 
present, would become XXTTL Or, by subtracting two u nits fro m all the 
XXIT and XXV would belong to 6 and 17, while XXIV, under 14 




.zz'.zr- 

oi XXIT £nd 



17 

would become XXII. This state of things will exist after the year 1900. 
The epacts of the present century are those of the foregoing series reduced 
by one. We have XXV and XXVI in the 6th and 17th years of the cycle, 
but XXIV does not occur at all. 

To meet the case in which XXIV and XXV are both present, it was pro- 
vided by Clavius that XXV should take, for the period over which this 
state of things extends, the vacant place of XXVI ; and this explains the 
introduction of the 25 in Arabic numerals, which stands by the side of 
XXVI in the table. The 25 is introduced into the full months as well as 
into the hollow months, though quite unnecessarily. By this contrivance, a 
separate day of the month is provided for every Paschal full moon. 

The practical advantage gained by this elaborate contrivance is exceed- 
ingly small. It prevents the possible occurrence of Easter on the same 
day of the month twice in the same cycle, when the Sunday letter happens 
to be C. It will be observed, from an inspection of the series of epacts just 
given, that if there are two in the cycle differing by a unit, the larger will 
arrive eleven years after the less. If XXIV and XXV are both present, 
therefore, the Golden Number corresponding to XXV must be greater than 
eleven. Eleven consecutive years will, three times out of four, embrace 
three leap years ; so that the Sunday Letter will movfe in this time, fourteen 
places ; or, in other words, two years distant from each other by an interval 
of eleven, will have the same Sunday Letter. If the> epact is XXV, the 
corresponding Paschal full moon will fall on the 18th day of April, to 
which, as the table shows, the letter C belongs. If, therefore, C is the Sun- 
day Letter, Easter will fall on the 25th April, which is the Sunday following. 
But the same will be true if the epact is XXIV. By changing XXV, how- 
ever, to 25 — that is, to XXVI — the Paschal full moon for this epact falls on 
Saturday, April l?th, and Easter is the next day, April 18th, while the epact 
XXIV still gives the 25th for Easter. It is doubtful whether the object se- 
cured by this elaborate contrivance was worth the trouble, especially when 
it is considered that Easter often unavoidably falls on the same day of the 
month within the same cycle, and is always liable to do so with certain 
Sunday Letters in years differing by an interval of jive or six or eleven.* 
Years differing by five have the same Sunday Letter if two leap years inter- 
vene ; and years differing by six similarly agree if but one leap year falls in 
the interval. Now, the years 1 and 7 of the cycle differ by 6, and have 
epacts differing by 6 ; and the years 2 and 7 differ by 5, and have epacts 
differing by 5. The same is true of other numbers of the series. There is, 
accordingly, always a liability to the recurrence of Easter on the same day, 
when the earliest of the two paschal full moons falls on Sunday or Monday. 
The chance of coincidence is much greater when years differ by eleven, since 
their epacts differ by only one. 

The duplication of epacts on the 5th of April has, as above remarked, 
the effect to make the distance between successive new moons only twenty- 
nine days in the limiting case ; while all those corresponding to greater epacts 
are distant by thirty. This occasions a little irregularity, which might have 
been avoided by making the duplication on a day later in the month ; in 
which case it would have affected no lunation regulating Easter. The plan 

* Thus, during the current cycle faster occurs on the same day in the following years 
differing by eleven, viz. : 

In 1863 and 1874 On the 5th of April. 

In 1865 and 1876 On the 16th of April. 

In 1866 ami 1877 On the 1st of April. 

In 1867 and 1878 On the 21st of April. 

In 1869 and 1880 . On the 28th of March. 

In the following differing by six, viz. : 

In 1869 and 1875 On the 28th of March. 

In 1873 and 1879 On the 13th of April. 

And in the following differing \>yflve, viz. : 

In 1S75 and 1880 On the 28th of March. 

2' 



18 

here suggested would have been attended with no practical consequence 
except that, in one combination of circumstances, it would have thrown 
Easter forward from the 19th to the 26th of April ; a matter of no moment, 
considering that the festival already occurs as late as the 25th. It would, 
doubtless, have been a great advantage had Easter been restricted to a much 
more limited range of movement, and not been tied to the moon at all ; but 
the saving of a day where the range still extends over thirty-five days, is an 
insignificant benefit. 

From what has been said, the break in the third of the general tables 
relating to Easter in the Prayer Book will be understood. We have seen 
that when the epact is XXV, and the year in the cycle is greater than 11, 
XXVI is taken instead of XXV, making Aprii 4th the day of new moon, 
instead of April 5th ; and April 17th the day of Paschal full moon, instead of 
April 18th. The break in the table occurs between the eleventh and twelfth 
years of the cycle, and all the later numbers stand opposite to April 17th. 

It has been stated that in the sixteenth century, after the reformation of 
the calendar had taken place, the epact 1 fell on the first year of the cycle. 
As the reformation, by the suppression of ten calendar days, had reduced 
all the epacts to a corresponding extent, it would appear that, previously to 
this time, the epact of the first year of the cycle had been 11. In point of 
fact, it was 8. Apparently it would have answered every practical purpose 
if the author of the new calendar had been content to take this fact as he 
found it. He thought proper, however, to make two or three arbitrary 
assumptions in order to connect his work with a much earlier epoch. As 
the lunar correction is governed by periods of twenty-five centuries, he took 
the year 1800 as the end of one of these periods. He assumed, accordingly, 
that the correction had been regularly applied in the years 1400, 1100, 
and 800.* He further assumed that, previously to the year 800, the epact 
was zero in the third year of the cycle (making it 8 in the first), and that 
this had been the case during the previous centuries, from the time of the 
council at Nicaea. The application of the lunar correction in the year 
800 was then supposed to increase the epact in the first year of the cycle 
to 9. A similar correction in 1100 was supposed again to make it 10 ; and 
a third in 1400 to make it 11 ; after which the suppression of the ten days, 
as above stated, reduced it to 1. By this artifice, the change in the epact 
made in 1582 was really only seven, while it was nominally ten. 

When the epact in the first year or last year of the cycle is known, that 
of any other year of the cycle may easily be calculated. As the increase, 
from the first year onward, is eleven days per annum, we have only to add 
to the epact of the first year, eleven, multiplied by the number of the year in 
the cycle, less one. Thus, if e / is the epact of the first year, and e that of 
the nth year, we shall have 

e = e' + (n—l) x 11= e' + n + 10 (n —1) — 1. 

Of course, if the result exceeds thirty, thirty, being an embolismic month, 
must be dropped. Putting e / = 1, as was true at the formation of the cal- 
endar, we shall have the simple formula, 

e = n + 10 (n — 1), 
or the epact is equal to the number of the year in the cycle, increased by ten 
times the number next less. The value of e f , however, became zero in 1700, 
and since that time the proper formula has been 

e = 11 (n - 1), 
or the epact is equal to eleven times the number next below that which ex- 

* To have completely eliminated the preexisting error of the epact, the imaginary cor- 
rection fhould have been applied also in the year 500, the ecclesiastical moon having been 
behind the astronomical mean moon more than four days. It was not thought advisable to 
do this, because it would have put the ecclesiastical moon, for half the time, earlier than 
the true moon, according to which latter the Jews regulated their month Niean, by direct 
observation. By keeping the calendar moon a day behind the astronomical mean moon, 
Clavius believed that lie had effectually provided against the possibility of the occurrence of 
Easter before the fourteenth of Nisan, or the Jewish Passover, in any year ; or before the 
true full moon of the Paschal month. 



19 

presses the place of the given year in the cycle. Or the former rule can be 
used, and the result diminished by one. In 1900 and after, the former 
rule may still be used, the result being diminished by tico. 

If z" be used to express the epact of the last year of the cycle, the value 
of e will be found by adding to e // , in the first place, twelve, which is the 
increase of the epact in passing from one cycle to another, and eleven for 
each succeeding vear. Thus : 

e = B " + 12 + 11 (n - 1) = e" + 11 n + 1. 

Thus, in the latter part of the sixteenth century, t" was 19. For the 
first vear of the cvcle, we shall find, putting n = 1, 

e = 19 + 11 + 1 = 31 ; or, dropping 30, e = 1. 

At present, e" = 18. Hence, for the present year, 1871, which is the 
tenth year of the cvcle, 

e = 18 + 11 x 10 + 1 = 129 ; excluding 30s, e = 9 

The equation, e = e // + 11 n + 1, may be put in this case : 

e = 18 + 11 + 11 (n - 1) + 1 = 11 (n - 1) + 30 = 11 {n - 1), 
which result agrees with that obtained before. 

When the epact is known, we find the day of new moon in March by 
looking into the ecclesiastical calendar given above, where this day stands • 
opposite the given epact. To find the day of full moon, we add thirteen, 
which carries us forward to the fourteenth day of the moon, so called. If the 
full moon happens earlier than the 21st of March, it cannot be the Paschal 
moon. In that case, we must, accordingly, look in April. 

A mode of proceeding which dispenses with the table is the following : 
The earliest day of March on which the Paschal full moon can fall is the 
21st. The new moon which corresponds to this happens on the 8th (21 — 
13 = 8), and the epact corresponding to the 8th is 23. Now, if the epact 23 
gives full moon the 21st, the epact 22 will give full moon the 22d, the epact 
21, full moon the 23d, and so on. In other words, as the epact diminishes, 
the date of full moon equally increases ; and the sum of this date and the 
epact leading to it is a constant sum. This constant is, of course, equal to 
21 + 23, or 44. 

The following rule, therefore, is general : Subtract the epact from 44, and 
the remainder is the date of full moon considered as a day of March. 

But if this remainder is less than 21 , the full moon so found cannot be 
the Paschal full moon. In this case, therefore, we must increase the con- 
stant 44 by 30 (the number of days intervening, as we have seen above, 
between the new moons of the early days of March and .those which follow 
them in April), giving us a new constant, 74 ; and from this, if, as before, we 
subtract the epact, we shall have the date of the Paschal full moon, which, 
though expressed as a day of March, will fall in April. 

Easter is the Sunday following this full moon. It is desirable, therefore, 
to know the calendar letter corresponding to the day on which the full 
moon falls. When this day is the 21st March, this letter is always C = 3. 
The epact which brings the full moon to March 21st is, as we have seen, 
= 23. If the full moon falls on the 22d, the letter becomes D = 4, and the 
epact recedes from 23 to 22 ; or, generally, as the value of the calendar 
letter belonging to the Paschal moon increases, in the same manner the 
value of the epact diminishes ; so that the sum of these two values is 
a constant quantity. This constant is, accordingly, = 23 + 3 = 26. And the 
rule for finding the calendar letter corresponding to the Paschal full moon is 
the following : 

Subtract the epact from 26. and the remainder (after suppressing sevens) 
is the value of the calendar letter sought. But, inasmuch as an epact 
greater than 23 requires, for finding the day of Paschal moon, a constant in- 
creased from 44 to 74, so here, for the higher epacts, the constant 26 must 
be raised to 56, by the same addition (of 30) as in the former case. Accord- 
ingly, when the epact exceeds 23. subtract it from 56, and the remainder 
(with sevens suppressed) will give the calendar letter sought. 

The Sunday following the Paschal full moon being Easter Sunday, we 



20 

may find liow many days later it falls by subtracting the moon's calendar 
letter, found as above, from the Sunday Letter for the year ; increasing this 
latter by seven, if necessary. The numerical difference thus found, added to 
the date of Paschal full moon, will give the day of the month upon which 
Easter falls. 

The system of epacts thus devised by Clavius, considered as a means of 
fixing the time of Easter and the other movable feasts of the Church, can- 
not be commended for its simplicity or its adaptation to popular use, or 
even its fitness to reach the popular intelligence. It would seem, indeed, 
as if it had not been intended to be popularly understood ; for certainly if 
the premeditated purpose of its author had been to devise a scheme for ren- 
dering a comparatively simple subject obscure, he could not have been 
more completely successful. 

The consideration of epacts keeps before the mind constantly the idea of 
a retrogradation of dates. This is by no means so simple of conception as 
that of an advance. But the annual backward movement of a new moon 
through the space of eleven days is practically equivalent to a forward 
movement of nineteen days, since nineteen is the complement of eleven to 
thirty, and thirty is the number of days in each embolismic month (the last 
excepted, which, because it is the last, affects no computation within the 
cycle). 

Again, the presentation continually before the mind, of the new moon, 
when we have nothing to do (ecclesiastically) with the new moon, except to 
take a day thirteen days later, is a source of quite unnecessary confusion. 
The direct and simple mode of finding the time of ecclesiastical full moon, 
which is what we do want to know — a mode, also, which has the merit of 
being easily intelligible — would have been, in the time of Clavius, and is 
now, to start from some known date of full moon, and to add nineteen days 
for each succeeding year (or eighteen for that last one in which the passage 
is made from the end of one cycle to the beginning of another), dropping out 
the thirties, as in the calculation of epacts. The result of this gives imme- 
diately the day itself required. The calculation is one which any person 
can make without difficulty ; but if it were made in advance, for a whole 
cycle, and the numbers tabulated, these numbers would occupy no more 
space than the epacts do now, and they would be the things wanted, and not 
intermediate instrumentalities for finding these things. Let us take, for 
instance, as a starting point, the last year of the cycle in our present cen- 
tury. Its epact, we, have seen, is 18. Subtracting this from 44, we have, as 
the date of Paschal full moon (which we may represent by P), the 26th of 
March. For the next following year there will be a full moon (as we are 
now passing from one cycle to the next) later by eighteen days. And for 
the years succeeding that, a full moon will follow nineteen days later each 
year. We shall then have the general expression, 
P = 26 + 18 + 19 (n - 1).* 
Or, P = 25 + 19 n = 10 + 15 + 15 n + 4 n. 
And therefore, P = 10 + 15 (n + 1) + 4 n. (A.) 

♦There is an analosry between this expression and one of the equations of Gauss, in 
his method for finding Easter, given by Delambre (Astronomic Theorique et Pratique, torn, 
i. p. 712), which is in the following form, viz. : 

d= ((19a + JO-5-30) r , 
in which a is the remainder left in dividing the given year of our Lord by 19, and is, there- 
fore, evidently = n — l. Though the formula is given without explanation, M is easily 
made out to be = 44 + 1 - e" + 18 - 22. The subscript r indicates that the remainder of the 
division by 30 only is to be taken. It is consequently apparent lhat d is ihe number of 
days from March 22 to the first Paschal day ; that is, to the first day on which, with this 
value of n, Easter can possibly fall ; which day is, of course, the first day after Paschal 
full moon. . ^ ... 

If I be put for the Calendar Letter of the fir?t Paschal day, and A for the Dominical 
Letter of the year, we shall have, for the time of Easter referred to March, 

~E=22 + d + A-l; 
to which seven must be added in case I exceeds A. 

As the Calendar Letter of March 22 is D = 4. the value of I is of course = d + 4 (suppress- 



21 

Now when n 4- 1 is even, 15 (n 4- 1) is a multiple of 30, and may be 
dropped. In this case, n itself is odd. Hence, when the number of the year 
in the cycle is odd, 

P = 10 + 4?i. 

For example, the year 1872 is the 11th of the cycle, and for that year, 
P = 10 4- 44 = 54 ; or, dropping 30, the Paschal full moon falls on the 24th 
of March. 

It may be remarked that, if n is equal to or greater than 15, we may drop 
15 from it before multiplying by 4 ; since 4 x 15 = 60, which is twice thirty. 
Thus, the year 1880 is the nineteenth of the cycle. Dropping 15 from 19, 
we have P = 10 + 4 x 4 = 26, which is the same as stated above for n = 19. 

ing sevens if necessary). The peculiarity of Gauss's method consists in his ingenious 
general expression for the value of A, which, by analyzing his formulae, we gather to be 

where Fis the given year of our Lord, and S is the Dominical Letter of some year less than 
T, of which the numerical value is a multiple of 28 ; which is, therefore, at the same time 
a bissextile year. That this expression is universally true, may be shown as follows : 

If Nbe taken to represent any number of years following the assumed bissextile year 
of which 6 is the Dominical Letter, we shall have, for the Dominical Letter of the JVTh year, 
the expression, . ,_ /K\ 

sevens being added in sufficient number to make the result positive. The subscript q de- 
notes that the quotient of the division only is to be taken. This may be written otherwise 
as follows : „ „ ,_ /N\ 

since the subtraction of a unit is equivalent, in its effect upon the Calendar Letter, to the 
addition of six units. The negative term here can have only the values — 1, — 2, — 3, — 4, 
etc. ; and these, without affecting the value of A, may be replaced by — 8, — 16, — 24, etc., 
since -8=-l-7; -16 =-2-2x7; -24 =-3-3x7, etc. 

Further, the positive term OJVmay be written = AN + 2N','- and when 2f= 4, i\T= 8, etc., 
it will have the values 4JV + 8, 4JV + 16, etc., so that, in these cases, the equation may be 
written, 

A = S + 4i^+S-8; A = 5 + 4i^+ 16-16, etc., 
[n other words, when If= 4, or any multiple of 4, the term 2iV and the negative term neu- 
tralize each other. Hence, generally, 

Moreover, when iVis a multiple of 7, the term 4-iVinay be suppressed without affecting 
A ; so that, for the case in which N is a multiple of both 4 and 7, the equation simplifies 

itself to A = 8 ; and 4JY is always equivalent to 4 ( — 1 r . Hence the general expression 

the sevens in the sum of these terms being suppressed. 

Consequently, /iV\ ft /iV\ ,, „ 

A-^5 + 4( T ) r + 2( T ) 7 .-(d + 4). 

° r ' A-* = S + 4(f) r + 2(f) r + 6tf + 24. 

Now, if we put F= 28m + iV, m being integral, we shall have, 

(f)r=(|)r;and(f) r =(f) r . 

Accordingly, if 28m be the number denoting the bissextile year of which $ is the Do- 
minical Letter, we may find the value of S, for any assumed value of m, by the ordinary 
methods. This value will be the same for any other multiple of 28; as, for instance, for 
28m' ; unless, between the values 28m and 28m', there intervene one or more non-bissex- 
tile centurial years 

Suppose we take m = 65 ; then 28m = 1S20 ; and for 1820, 5 = A = 1. Then, 

A-J = l+4(-^) r + 2(j) r + 6d + 24; 
Or, uniting the numerical terms and suppressing sevens, 

A -' = 4 (fV +2 (f> + 6c * + 4 - 

If m = 68. 28m =1904; and 5 = 2? = 2 ; whence the final term of the foregoing expres- 
sion becomes 5 for yeara between 1899 and 2099, exclusive of the first of these years, and 
inclusive of the last. In Gauss's formulae this variable numerical term is represented gen- 
erally by the letter -ZV. for which we have found a different provisional use above. 

An improvement on Gauss's method was suggested in the year 1817, by Father Ciccolini, 



for A is this, viz. : „ M /jy"> 

A = " 



22 

But if 71 + 1 is odd (in which case n itself must be even) then the for- 
mula (A) foregoing may be written, 

P = 10 + 15 + 15 n + 4 n. 
Or (15 n being a multiple of 30), 

P = 10 + 15 + 4 n = 25 + 4 n. 

And here, as before, if n equals or exceeds 15, 15 may be dropped before 
multiplying. 

These formulae embrace the simple rules given in the first part of this 
paper for finding the time of Paschal full moon. If we had taken the epact 
of the last year of the cycle as it stood before 1700 (it was 19 instead of 18), 
we should have found P = 25 (instead of 26) for that year ; and the result 
would have been to make the constants (10 and 25) in the final formulae a 
unit less. Thus, for the 16th and 17th centuries* 

When n is odd, P = 9 + 4 n. (B.) 

And when n is even, P = 24 + 4 n. (B'.) 

During these centuries the secular correction in the General Table II of 
the Prayer Book was zero. The value of this correction, as it accrues, is to 
be added to the constants foregoing ; and we see now why the number nine, 
subtracted from the lesser of these constants, will always (as stated in the 
foregoing pages) leave as a remainder the secular correction corresponding 
to the century found in the table of the Prayer Book just referred to. 

Clavius laboriously reduced his whole system, including the necessary 
secular corrections for long periods, to tabular form. Delambre, in the first 
volume of his " History of Modern Astronomy," and De Morgan, in a paper 
contributed to the " Companion to the British Almanac " for 1845, have 
given rules for the calculation of all the elements which enter into the de- 
termination of Easter. The formulae devised by the celebrated mathema- 
tician Gauss for the same purpose, with the modification proposed by Cicco- 
lini, have been noticed in the note referred to on page -So©-. Delambre also 
presented his rules in the form of algebraic expressions. The formula 

an ecclesiastic of Rome, the object of which was to render the term iVa constant. It will 
be observed that the variability of N arises from the occurrence of non-bissextile centurial 
years. The most general form, therefore, of stating the value of A, or that which makes it 
true for all centuries, is the following : 



*=»-r-(fV ff -(?)«- 



in which C stands for the number of complete centuries, and Y, as before, stands for the 
entire number of years in the given year of our Lord. For the Julian Calendar, or old 
style, the value of A is always given by the first three terms of this expression. The final 
two, which correct for the omission of three leap years in four centuries, explain them- 
selves. We have seen how to dispose of the terms containing Y. As for those dependent 
on C we add, in the first place, 7(7. These terms then become, 

8<r-(£) ff =6cr + se'-(£) ff =6(f) r+S (£.) r . 

Hence, the value of A becomes, 

A = S + 4(|) r+9 (f) r+ 6(f) p+S (f) r . 

If, now, 5 be the Dominical Letter of the year zero, supposing the Gregorian computa- 
tion to be extended so far back, the foregoing formula will give the Dominical Letter of any 
other year in any century, 8 remaining constant. This constant value of 8 is evidently that 
of every bissextile centurial year ; and this we have already seen to be = 1. 

Gauss uses the letter e to stand for the value of A — J, with the sevens suppressed. 
Hence, for Easter, as referred to March, we have, 

E = 22 + d + e. 

From the manner in which the value of M, in the expression first cited above, is ob- 
tained, it is manifest that this value will be affected by the secular correction of the epact, 
increasing as the epact diminishes, and vice versa. It will also be understood, alter what 
has been said of the doubled epacts, XXIV and XXV, that when d is found = 29, it must 
always be taken = 28 ; and that when d is by formula = 28, it must be taken = 27 in case 
the Golden Number exceeds eleven. 

* As stated in a former note, the constant term in finding P before the reformation of 
the calendar was two. Accordingly, for old style dates, we have the equations following : 
Whon n is odd, P = 2 + 4*. 
Win n n is even. P = 17 + 4 ti. 



23 

above given for the epact in the 16th century, viz., e = n + 10 (n ■— 1), was 
first stated by him. 

Considering, then, that the epact, by the effect of the solar correction, is 
diminished three days in every 400 years, counted after 1600, Delambre pro- 
ceeded to give the epact as affected by this correction thus (putting C for the 
number of complete centuries) 

e = n + 10 (n - 1) - f (C - 16) = n + 10 (n - 1) - (C - 16) + \ (C - 16), 
the integral value of the fractional expression only being used. Consider* 
ing, also, that the lunar correction increases the epact one day every three 
hundred years, and that it was first applied in 1800, he finds that the effect 
of this correction, without regarding the irregularity which occurs at the 
end of every 25 years, may be expressed by the formula (putting c for cor- 
rection), 

C-15. 

c= -ir-' 

the integral value of the fraction only being used. Now, the eighth correc- 
tion after 1800 (which centurial year marks the end of the period of sup- 
posed or imaginary corrections preceding the actual introduction of the 
reformation) would fall, according to this formula, on the centurial year 
4200 (the twenty-fourth after 1800), whereas the theory requires that it 
should be deferred to the twenty-fifth ; that is, to the centurial year 4300. 
In like manner, theory requires that the eighth correction after 4300 should 
be deferred to the twenty-fifth centurial year following that date, instead of 
falling upon the twenty-fourth. 

Hence, it is necessary to place, in the numerator of the fraction express- 
ing the lunar correction, as above, a negative fractional term which shall 
increase with the progress of time, and shall gain a unit in twenty-five 
years, becoming integral first in 4200. The effect of this term will be to 
reduce 42 to 41, and thus defer for a century the advance of the value of c. 
Such a term is 

C-17, 

• 25 

42 — 17 
Since 25 + 17 = 42, and — -— = 1. 
25 
The complete expression for the value of c, therefore, is 

__ C - 17 

C - 15 25 : 

c — — 



3 

and as this goes to increase the epact, the general expression for the epact 
becomes 

C-17 



/ /'= n + 10 (n - 1) - (C — 16) + — j— + > 

an expression sufficiently formidable. 

But, though this degree of complication is necessary to the complete 
algebraic statement of all the conditions affecting the value of the quantity 
sought, yet, for practical purposes, the irregularity of long period may be 
disregarded, a special provision being made for it, to be separately applied. 
Then, if we put s to represent the combined effect of the secular corrections 
due to both sun and moon, we may state this effect as follows : 

,n -.m C-16 C-15. 

s= -(C-16) + — — + — — - 

Or, s=-C + 16 + iC-4 + i-C-5=-C + iC + iC + 7; 
in which the three terms containing C must not be united, since the integral 
values of the fractional terms only are employed. 

This is the secular correction of the epact. But, inasmuch as the date of 
full mOon advances as the epact diminishes, and vice versa, by the same 



24 

amount, this, with signs reversed, is also the secular correction of the con- 
stant in equation (B) which gives the value of P. To make that equation 
general, introduce S (= — s) to represent the correction, and we shall have, 

P = 9 + S + 4n. 
Or, substituting the value of — s in place of S, 

P = 9 + C-iC--LC-7 + 4tt = C-iC-iC + 2 + 4 7i. 
And this shows us that the constant for the century is found, as the rule 
given earlier in this paper states, by subtracting from the centurial number 
its fourth part and its third part successively, and increasing the result by 
two. From what has just been said, we see also the reason for the special 
rules in relation to the centurial years 4200, 6700, 9200, etc., which follow 
the general rule referred to. 

V. The fifth rule relates to finding the date of Easter Sunday, after that 
of the Paschal full moon has been ascertained. This will not require many 
words. It will be observed that the calendar letter, Gr, nearest preceding 
the earliest possible date of Easter (March 22), corresponds to the 18th day 
of March. If, then, to eighteen we add the numerical value of the Domini- 
cal Letter of the year, the sum will be the date of a Sunday. If this date 
exceeds that of Paschal full moon, it will be the date of Easter Sunday. If 
not, seven must be added often enough to obtain a date exceeding that of the 
Paschal full moon, each addition of seven giving, of course, a later Sunday. 
Thus the rule is justified. 

These explanations have been protracted beyond the original intention. 
It has been found difficult, in less space, to present clearly the reasons on 
which the new rules are founded. That these rules are preferable, in 
respect to simplicity and facility of application, to any which have been 
heretofore proposed, will probably be admitted by any one who will take 
the trouble to compare them with those of Gauss and Ciccolini, or with 
those laid down by De Morgan in his article already referred to, which are 
probably the best hitherto published, or, finally, with the cumbrous formulae 
of Delambre. 

I have the honor to be, Reverend and dear Sir, 

Your obedient servant, # 

The Rev. B. I. Haight, S.T.D., LL.D. F. A. P. BARXARD. 



V. 



TABLE REFERRED TO IX RESOLUTION 3. 



TEAKS OF OTTE LORD. 


GOLDEN 2TCMBER. 


THE EPACT. 


SUXDAY LETTER. 


EASTER-DAT. 


1881.." 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

IS 

14 

15 

16 

17 

18 

19 



11 

22 
3 

14 
25 

6 
17 
28 

9 
20 

1 
12 
23 

4 
15 
26 

7 
18 


B 

A 

G 
FE 

D 

C 

B 
AG 

F 

E 

D 
CB 

A 

G 

F 
ED 

C 

B 

A 


April 17. 

9. 

March 25. 
April 13. 


1882 

1883 

1S84 


18&T 


1886 


" 25. 


1887 


10. 


1888 


1. 


1889 


" 21. 


1890 

1891 


6. 
March 29. 


1892 


April 17. 
2. 


1893 


1694 


March 25. 


1895 

1896 


April 14. 


1897 


" 18. 


1S98 


" 10. 


1899 


2. 







ADDENDA. 



I. Rules for the Sundays after Trinity. 
1. To determine on what day any given Sunday after Trinity will fall 

Write in their order, as below, the names of the several months, from 
May to November, inclusive. Immediately under these, severally, write 
the terms of a numerical series beginning with 1, and increasing by the 
successive differences 5, 4, 4, 5, 4, 4 ; differences easily remembered by their 
symmetry of arrangement. Under these, again, write the three even num- 
bers less than seven, the three odd numbers less than seven, and the zero, 
thus : 2, 6, 4, 1, 5, 3, — where the terms of the two triplets of significant 
figures, though not in regular arithmetical progression, are symmetrically 
arranged. These last numbers are to be called the indices of those above 
them. The result will then be as follows : 



Months, 


May, 


June, 


July, 


Aug., 


Sept., 


Oct. 


Nov 


Sundays, 


1 


6 


10 


14 


19 


23 


27 


Indices, 


2 


6 


4 


1 


5 


3 






This little table, when once formed, may be preserved for permanent 
use ; but it is so simple that it may be reconstructed, without much trouble, 
for each occasion. 

Let, now, E stand for the date of Easter (referred to March) ; S, for the 
number of the Sunday of which the date is required ; T, for the term in 
the Sunday series which is next less than 8; and i, for the index of this 
term. Find, then, a numerical term, a, by the following equation : 
a = 7(S-T) + i. 

Then, d, the required date of the given Sunday, as referred to the month 
standing over the term T in the small table above, will be found by this 
formula : 

d = E + a. 

Thus, in 1871, when Easter is 9th April = 40th March, we have, for the 
twenty-first Sunday after Trinity, 19 as the Sunday term, with 5 as its index. 

Then, 8- T= 21 - 19 = 2 ; and 7 x 2 + 5 = 19 = a. 
Whence E + a = 40 + 19 = 59th September = 29th October = date required. 
To show the operation of the rule in extreme cases, take the year 1818, 
when Easter was March 22 ; and the year 1886, when Easter will be April 
25 = March 56. 

In 1818, for the 18th Sunday after Trinity, we have T— 14, and i = 1. 
Then, 

8- T= 18 -14 = 4; and 7x4 + 1=29 = a. 
Whence E + a = 22 + 29 = 51st August = 20th September = date required. 
In 1886, for the 5th Sunday after Trinity, T— 1 and i = 2. 

8-T=5- 1=4; and 7x4 + 2 = 30 = a. 
E + a = 56 + 30 = 86th May = 55th June = 25th July = required date. 
In case 8= T, the solution is very simple ; since d = E + i, immediately. 
Thus, in 1886, the 10th Sunday after Trinity gives, 

E + i = 56 + 4 = 60th July = 29th August = required date. 



26 



2. Given the date of any Sunday after Trinity, to determine its number. 

In this case, if d is not greater than E (referred to March), it must be 
increased by adding to it the number of days in the month preceding (or, if 
necessary,in the two months preceding), when the following will be true, viz. : 

The index, i, points out, of course, the Sunday term, T. Then, if we 
put _ZV for the number sought, and take n, an additional term found from 
the following, viz., 

we shall have finally, 

JS r =T+n. 
Thus, in 1818, the 19th of July = 49th of June was a Sunday, and 

Id - E\ /49 - 22\ /27\ . . 3 m n 

Then!^-— — ) = (^f\ = n = 3 ; and N= r+7i = 6 + 3 = 9th Sunday 



Trinity. 



after Trinity. 

In 1886 the 21st November will be a Sunday. As Easter, in that year, 
will be April 25 = March 56, we must add 61 days for the two months pre- 
ceding November, making d = 82d September. Then, 

/d-E\ /82-56\ /26\ B . J m ,„ 
^__j r = (__j r = ^ r= 5 = *; and 7- 19. 

Also, I— J =n = d; and JV= T+ n = 19 + 3 = 22d Sunday after Trinity. 

With a little practice, this method may be employed mentally. For this 
purpose, the relation of the months to the Sunday terms and their indices 
may be fixed in the mind, by remembering that the middle month is August, 
and that the corresponding Sunday term, 14, is one half the sum of the ex- 
tremes, since 1 + 27 = 28, and 28 = 2 x 14. The middle index is also unity. 

II. General Rules for Placino Days op the Month in the Calen- 
dar op the Week, and the Contrary. 

1. To determine the day of the week on which any given day of any mofith will fall. 

The Sunday Letter for the year is presumed to be known, or to have 
been found by the methods already given. The year is divided into four 
quarters, as usual, beginning with January, April, July, and October. 

The months from April to October form what is called the summer half ; 
tho.-e from October to April, the winter half. 

The days of the week are numbered from one to seven. During the 
summer half, Sunday is the first day of the week, and Saturday the 
seventh. During the winter half, Monday is the first day, and Sunday the 
seventh. 

The difference between the value of the Sunday Letter and seven is 
called the complement of the Sunday Letter. 

The number of days by which any month exceeds four weeks — twenty- 
eight days — is called its excess. 

The number of days by which any month falls short of five weeks — 
thirty-five days — is called its supplement. 

The excess will always be either 2 or 3 ; the supplement will always be 
either 5 or 4. The excess and the supplement of February will always be 
zero ; the added day of leap year being compensated by the change of Sun- 
day Letter in the succeeding months. 



27 

Put, now, d = day of month ; 6 = day of week ; k = Sunday Letter 
(ypa.fj.fia nvpiandv) ; k = complement of this letter ; e = the excess of the 
month, or sum of the excesses of the months, preceding the given month 
in the same quarter ; and e = complement oi e = supplement or sum of 
supplements of the same months. Then this equation will be true : 



( d + e + k \ 

\ 7 )r> 

id 6 = 7, equal 



it being understood that 6 = 0, and 6 = 7, equally denote the last, or seventh 
day of the week. 

Take, for example, July 4, 1776. In this year k = F= 6, and& = 1. July 
being the first month in the quarter, e = 0. Then, 

6 = i = ) = 5th day of the week = Thursday. 

The inauguration of Gen. Washington as first President of the United 
States took place on March 4, 1789. For this year, k = D = A; k = d; and 
e = S. Hence, 

6 = I = J = 3d day of the week = Wednesday ; 

the first day during the winter half being Monday. 

For the months of January and February in leap years, care must be 
taken to use the Sunday Letter belonging to those months. The rule for 
Sunday Letter heretofore given, finds, in leap year, the letter for March and 
the subsequent months. The letter for the two months preceding is one 
place more advanced in alphabetic order, or one unit higher in value. Thus, 
in 1776, k = G = 7 for January and February. 

Again, the birth of Gen. Washington occurred on the 22d February, 
1732, new style, in which year the Sunday Letter was £J=5 after February ; 
and F= 6 in February and January. Hence, k = l, and e = 3, so that 

6 = i - 1 = 5th day of the week = Friday. 

Washington's birth occurred, however, before the adoption of the Gre- 
gorian reckoning ; and in old style it is said to have taken place on the 11th 
February, 1731 ; the beginning of the year having then been fixed at March 
25. In the application of these formulae to dates in old style, it is necessary 
to add a unit to the number of the year for dates from January 1 to March 
24, inclusive, before finding the Sunday Letter. Afterward, the process is 
the same as for new style. 

Thus, stating Washington's birthday as February 11, 1732 (old style cor- 
rected for this purpose), we shall have k = A = 1, for the months following 
February, and k = B = 2 for February and January ; whence k = 5, and 

6 = [ = 1 = 5th day = Friday, as before . 

This formula may be expressed in words as follows : 

To the month-date add the excesses of the months preceding (if any) in the same 
quarter, and the complement of the Sunday Letteb. The excess of sevens in the 
sum is the value of the day of the week. 

2. To determine the days in any month which will fall on a given day of the week. 

For this case we have a formula which will easily be borne in mind from 
its analogy to the foregoing, viz. : 

The date thus determined will be the earliest in the month which can 
fall on the given day of the week. By adding sevens to this, others may 
be obtained — three always, and sometimes four. 

As an example of the use of this formula, take the following : The next 
meeting of the General Convention of the Church in the United States will 



28 

take place on the first Wednesday in October, 1874. On what day of the 
month will this meeting occur ? 

In 1874, k — D = 4 ; and in October, since this is a month of the winter 
half, Wednesday is the third day of the week. Also, since October is the 
first month of the quarter, e = 0. Then, 



-F^VF^-V* 



signifying that Wednesday is the last day of September ; so that the first 
Wednesday in October is October 7th. 

The meeting of Congress takes place on the first Monday in December, 
annually. In 1872 this day will be the second of the month; for, since 
k : = 6 ; £ = 9 ; and Monday is the 1st day of the week, 

d = ( =— — I = 2d day of the month. 

The annual Commencement at Columbia College is held on the last 
Wednesday in June. Wednesday, during the summer half, is the fourth 
day of the week. In 1872, k = 6 ; and for June, e = 9 ; whence, - 

d = I 1 = 5th day of June for the 1st Wednesday ; 

and 5 + 7 + 7 + 7 = 26th day of June for the last Wednesday = date re- 
quired. 

This formula may thus be reduced to words : 

To the week-date add the supplements of the months preceding (if any) in the 
same quarter, and the value of the Sunday Letter. The excess of sevens in the 
sum is the earliest month-date falling on the given day of the week. 

The following formulae may be used without first finding the Sunday 
Letter : 

Put q — quaternial = number of leap years (or of fours) in the incom- 
plete century ; r = residual = excess of fours in the same ; p = 7 — r = 
complement of residual; r'= century-residual = excess of fours in the num- 
ber of complete centuries. Then, for day of week, 
■■ fd + e + r + 5(q + r') — 1\ 

e= \ t y 

and, for day of month, 



_ ( 6 + e + p + 2(q + r') + 1 \ 



In leap years, for dates in January and February (only), the numerical 
term in the foregoing formulae should be 2 instead of 1 ; for the remaining 
months, from March to December inclusive, the terin,l is correct as it stands. 

The foregoing are true for the Gregorian reckoning. The following are 
applicable to the Julian : 

Put v = excess (vnepoxv) for the century ; i. e., the number of complete 
centuries, with sevens suppressed ; u — 7 — v % complement of excess ; then, 



_f d + e + r + 5g + u — 3 \ 
-[ 7 /r ;an 

/ d + e + p + 2q + v+3 \ 



7 Jr 

In leap years, for dates in January and February (only), the numerical 
term in these formulae should be 4 instead of 3. 

In British (and British colonial) old style dates falling in January, Feb- 
ruary, or March (to March 24, inclusive), the number of the year should be 
increased by unity before applying the formulae. 

If, in the use of any of these formulae, a negative value should be 
obtained, it must be made positive by adding seven. 



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